\[{H^{\pm}_{\ell}}\left(\eta,\rho\right)=(\mp\mathrm{i})^{\ell}e^{(\pi\eta/2)\pm%
\mathrm{i}{\sigma_{\ell}}\left(\eta\right)}W_{\mp\mathrm{i}\eta,\ell+\frac{1}{%
2}}\left(\mp 2\mathrm{i}\rho\right),\]